Selectivity of Frequency

If we now plot the magnitude of the current $I = E/Z_T$ versus frequency for a fixed applied voltage E, we obtain the curve shown in Fig. 1, which rises from zero to a maximum value of $E/R$ (where $Z_T$ is a minimum) and then drops toward zero (as $Z_T$ increases) at a slower rate than it rose to its peak value.
Current versus frequency for the series resonant circuit
Fig. 1: Current versus frequency for the series resonant circuit.
The curve is actually the inverse of the impedance-versus-frequency curve. Since the $Z_T$ curve is not absolutely symmetrical about the resonant frequency, the curve of the current versus frequency has the same property.
There is a definite range of frequencies at which the current is near its maximum value and the impedance is at a minimum. Those frequencies corresponding to $0.707$ of the maximum current are called the band frequencies, cutoff frequencies, or half-power frequencies. They are indicated by $f_1$ and $f_2$ in Fig. 1.
The range of frequencies between the two is referred to as the bandwidth (abbreviated BW) of the resonant circuit.
Half-power frequencies are those frequencies at which the power delivered is one-half that delivered at the resonant frequency; that is, $$ \bbox[10px,border:1px solid grey]{P_{HPF} = {1\over 2} P_{max}} \tag{1}$$
The above condition is derived using the fact that $$ P_{max} = I^2_{max}R$$ and $$ P_{HPF} = I^2R = (0.707 I_{max})^2R\\ =(0.5)I^2_{max}R = {1\over 2} P_{max} $$
Since the resonant circuit is adjusted to select a band of frequencies, the curve of Fig. 1 is called the selectivity curve. The term is derived from the fact that one must be selective in choosing the frequency to ensure that it is in the bandwidth.
The smaller the bandwidth, the higher the selectivity. The shape of the curve, as shown in Fig. 2, depends on each element of the series R-L-C circuit. If the resistance is made smaller with a fixed inductance and capacitance, the bandwidth decreases and the selectivity increases. Similarly, if the ratio L/C increases with fixed resistance, the bandwidth again decreases with an increase in selectivity.
Effect of R, L, and C on the selectivity curve
for the series resonant circuit.
Fig. 2: Effect of R, L, and C on the selectivity curve for the series resonant circuit.
In terms of $Q_s$, if $R$ is larger for the same $X_L$, then $Q_s$ is less, as determined by the equation $Q_s = w_s L/R$.
A small $Q_s$, therefore, is associated with a resonant curve having a large bandwidth and a small selectivity, while a large Qs indicates the opposite.
For circuits where $Q_s \geq 10$, a widely accepted approximation is that the resonant frequency bisects the bandwidth and that the resonant curve is symmetrical about the resonant frequency.
These conditions are shown in Fig. 3, indicating that the cutoff frequencies are then equidistant from the resonant frequency.
Effect of R, L, and C on the selectivity curve
for the series resonant circuit.
Fig. 3: Approximate series resonance curve for $Q_s \geq 10$
For any $Q_s$, the preceding is not true. The cutoff frequencies $f_1$ and $f_2$ can be found for the general case (any $Q_s$) by first employing the fact that a drop in current to $0.707$ of its resonant value corresponds to an increase in impedance equal to $1/0.707 = \sqrt{2}$ times the resonant value, which is R.
Substituting $\sqrt{2} R$ into the equation for the magnitude of $Z_T$, we find that $$ Z_T = \sqrt{R^2 + (X_L - X_C)^2}$$ becomes $$ \sqrt{2} R = \sqrt{R^2 + (X_L - X_C)^2}$$ or, squaring both sides, that $$ 2R^2 = R^2 + (X_L - X_C)^2$$ and $$ R^2 = (X_L - X_C)^2$$ Taking the square root of both sides gives us $$ \bbox[10px,border:1px solid grey]{R = X_L - X_C }\tag{2} $$ or $$ R - X_L + X_C = 0$$ Substituting $wL$ for $X_L$ and $1/wC$ for $X_C$ and bringing both quantities to the left of the equal sign, we have $$ R - wL + {1 /over wC} = 0$$ or $$ Rw - w^2L + {1 /over C} = 0$$ which can be written $$ w^2 - {R \over L} w - {1 /over LC} = 0$$ Solving the quadratic, we have $$w = {-(-R/L) \pm \sqrt{[-(R/L)]^2 -[-(4/LC)]} \over 2}$$ and $$\bbox[10px,border:1px solid grey]{w = {R \over 2L} \pm {1 \over 2} \sqrt{{R^2 \over L^2}+{4 \over LC}}} \tag{3}$$ Let us first consider the case where $X_L > X_C$, which relates to $f_2$ or $w_2$. Also Eq.(2) relates $R = X_L-X_C$ to be a +ive value ($R=X$). Hence Eq.(3) becomes, $$w_2 = {R \over 2L} \pm {1 \over 2} \sqrt{{R^2 \over L^2}+{4 \over LC}}$$ with $$\bbox[10px,border:1px solid grey]{f_2 = {1 \over 2 \pi}[{R \over 2L} + {1 \over 2} \sqrt{{R^2 \over L^2}+{4 \over LC}}]} \, (Hz) \tag{4}$$ The negative sign in front of the second factor was dropped because ${1 \over 2} \sqrt{{R^2 \over L^2}+{4 \over LC}}$ is always greater than $R/(2L)$. If it were not dropped, there would be a negative solution for the radian frequency w.
If we repeat the same procedure for $X_C > X_L$, which relates to $w_1$ or $f_1$ such that $Z_T = \sqrt{R^2 + (X_L - X_C)^2}$. Also $R = X_L-X_C$ in Eq. (2) results in a -ive value ($R=-X$), the solution $f_1$ becomes $$\bbox[10px,border:1px solid grey]{f_1 = {1 \over 2 \pi}[{-R \over 2L} +{1 \over 2} \sqrt{{R^2 \over L^2}+{4 \over LC}}]} \, (Hz) \tag{5}$$ The bandwidth (BW) is $$BW = f_2 - f_1 = \text{Eq. (4)} \, - \text{Eq. (5)}$$ and $$BW = f_2 - f_1 ={ R \over 2 \pi L}$$ Substituting $R/L = w_s/Q_s$ from $Q_s = w_sL/R$ and $1/2 \pi = f_s /w_s$ from $w_s = 2 \pi f_s$ gives us $$BW ={ R \over 2 \pi L} = ({ 1\over 2 \pi})({R \over L}) = ({ f_s \over w_s})({w_s\over Q_s}) $$ or $$\bbox[10px,border:1px solid grey]{BW ={ f_s \over Q_s} } \tag{6}$$ which is a very convenient form since it relates the bandwidth to the $Q_s$ of the circuit. As mentioned earlier, Equation (6) verifies that the larger the $Q_s$, the smaller the bandwidth, and vice versa. Written in a slightly different form, Equation (20.21) becomes $$\bbox[10px,border:1px solid grey]{ {f_2 - f_1 \over f_s ={ 1\over Q_s} } \tag{7}$$ The ratio (${f_2 - f_1 \over f_s}$ is sometimes called the fractional bandwidth, providing an indication of the width of the bandwidth compared to the resonant frequency.
It can also be shown through mathematical manipulations of the pertinent equations that the resonant frequency is related to the geometric mean of the band frequencies; that is, $$\bbox[10px,border:1px solid grey]{f_s = sqrt{f_1f_2}}$$ Geometric mean of any numbers can be found by using formula $$ G.M = \prod_{i=1}^n X_i$$